陶哲轩博客写数学问题时，通常先把对象定义清楚，再给直觉、反例和证明轮廓。把「Hilbert’s seventh problem, and powers of 2 and 3」改写成可阅读的中文笔记，重点是：问题在问什么、已知到哪一步、下一步最容易走偏在哪。原站广告、分享条和导航已去掉。

问题在问什么

I’ve been focusing my blog posts recently on the mathematics around Hilbert’s fifth problem (is every locally Euclidean group a Lie group?), but today, I will be discussing another of Hilbert’s problems, namely Hilbert’s seventh problem , on the transcendence of powers of two algebraic numbers . (I am not randomly going through Hilbert’s list , by the way; I hope to explain my interest in the seventh problem in a later post.) This problem was famously solved by Gelfond and Sc

Theorem 1 (Gelfond-Schneider theorem) Let be algebraic numbers, with and irrational. Then (any of the values of the possibly multi-valued expression) is transcendental.

已知结果和反例

For sake of simplifying the discussion, let us focus on just one specific consequence of this theorem:

Proof: If not, one could obtain a contradiction to the Gelfond-Schneider theorem by setting and . (Note that is clearly irrational, since for any integers with positive.)

证明或构造的主线

In the 1960s, Alan Baker established a major generalisation of the Gelfond-Schneider theorem known as Baker’s theorem , as part of his work in transcendence theory that later earned him a Fields Medal. Among other things, this theorem provided explicit quantitative bounds on exactly how transcendental quantities such as were. In particular, it gave a strong bound on how irrational such quantities were (i.e. how easily they were approximable by rationals). Here, in particular,

Proposition 3 (Special case of Baker’s theorem) For any integers with positive, one has

阅读时建议盯住的点

for some absolute (and effectively computable) constants .

This theorem may be compared with (the easily proved) Liouville’s theorem on diophantine approximation, which asserts that if is an irrational algebraic number of degree , then

值得单独记下的条目

• A nonstandard number is said to be of polynomial size if one has for some standard .
• A nonstandard number is said to be of polylogarithmic size if one has for some standard .
• A nonstandard number is said to be of quasipolynomial size if one has for some standard .
• A nonstandard number is said to be quasiexponentially small if one has for every standard .
• Given two nonstandard numbers with non-negative, we write or if for some standard . We write or if we have for all standard .
• The sum, product, or difference of two quantities of a given size (polynomial, polylogarithmic, quasipolynomial, or quasiexponentially small) remains of that given size (i.e. each size range forms a ring).
• If , and is of a given size, then is also of that size.
• If is of quasipolynomial size and is of polylogarithmic size, then is of quasipolynomial size, and (if is a natural number) is also of quasipolynomial size.

阅读和落地时建议先做的 5 件事

1. 用自己的语言重写定义和结论，不看原文能不能说清对象是什么。
2. 找一个最小反例或边界情形，确认假设少一条会怎样。
3. 把证明拆成可独立检验的引理，每步只保留一个新想法。
4. 若涉及计算或形式化，先写可复现的小例子，再谈一般情形。
5. 记下尚未解决的缺口：缺估计、缺构造，还是缺正确的范畴。

和智能体、形式化工具怎么接

龙虾PRO做 OpenClaw 落地时，数学笔记最有用的部分往往是「可检验的步骤」：定义、反例、引理边界。智能体适合帮忙展开计算和检索，不适合代替你决定哪条假设能扔。

本文侧重全链路风控方法论。落地时请用自身业务单据做回放验证，不要把示例阈值直接当生产策略。 相关：风控体检 · 方案资源